| import OAI.Algebra.Finitistic.Bar.CountRetracts |
| import OAI.Algebra.Finitistic.Bar.DirectSumExactness |
| import Mathlib.Algebra.Homology.DerivedCategory.HomologySequence |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex TrivSqZeroExt |
| noncomputable section |
| namespace DerivedBar |
| universe u v w z |
|
|
| theorem nonzero_of_functor_retract {C : Type u} [Category.{v} C] |
| {E : Type w} [Category.{z} E] [HasZeroMorphisms E] |
| (F : C ⥤ E) {V W : C} (i : V ⟶ W) (p : W ⟶ V) (hip : i ≫ p = 𝟙 V) |
| (h : ¬ IsZero (F.obj V)) : ¬ IsZero (F.obj W) := by |
| intro hz |
| apply h |
| rw [IsZero.iff_id_eq_zero] |
| have hm := congrArg F.map hip |
| rw [F.map_comp, F.map_id] at hm |
| rw [← hm, hz.eq_of_tgt (F.map i) 0, zero_comp] |
|
|
| variable (D : Type u) [Ring D] |
| local instance countHomologyDetectionHasDerivedCategory : |
| HasDerivedCategory.{u+1} (ModuleCat.{u} D) := |
| HasDerivedCategory.standard (ModuleCat.{u} D) |
|
|
| /-- Exact comparison of integer derived homology and the original chain degree. -/ |
| def integerQHomologyIso (K : ChainComplex (ModuleCat.{u} D) ℕ) (n : ℕ) : |
| (DerivedCategory.homologyFunctor (ModuleCat.{u} D) (-(n:ℤ))).obj ((integerQ D).obj K) ≅ |
| K.homology n := |
| (DerivedCategory.homologyFunctorFactors (ModuleCat.{u} D) (-(n:ℤ))).app |
| ((integerExtension D).obj K) ≪≫ integerExtensionHomologyIso D K n |
| end DerivedBar |
| namespace DerivedBar.ParityBridge |
| universe u |
| variable (k D X : Type u) [Field k] [Ring D] [Algebra k D] |
| [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X] |
| [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X] |
| local instance (V : ModuleCat D) : Module k V := BimoduleTensor.scalarModule k D V |
| local instance (V : ModuleCat D) : IsScalarTower k D V := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| variable (N : ModuleCat D) |
| local notation "A" => TrivSqZeroExt D X |
| local instance : Module A N := inflatedModule D X N |
| local instance : IsScalarTower k A N := IsScalarTower.of_algebraMap_smul |
| (fun a v => IsScalarTower.algebraMap_smul D a v) |
| local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} D) := |
| HasDerivedCategory.standard (ModuleCat.{u} D) |
| local notation "S" => SquareZero.twoSidedWeightComplex k D X N (inflated_ideal_zero D X N) |
| local notation "T" => tensorBarComplex k A D N (fstHom k D X) |
|
|
| /-- Uniform vanishing of the original count homologies implies vanishing of |
| the entire actual balanced tensor bar, including its infinite count family. -/ |
| theorem tensorBar_homology_zero_of_counts (n : ℕ) (h : ∀ r, IsZero ((S r).homology n)) : |
| IsZero ((T).homology n) := |
| (directSumComplex_homology_zero D S n h).of_iso |
| ((HomologicalComplex.homologyFunctor (ModuleCat D) (.down ℕ) n).mapIso |
| (SquareZero.tensorCountSplitting k D X N (inflated_ideal_zero D X N))) |
|
|
| theorem tensorBar_homology_nonzero_of_count (r n : ℕ) (h : ¬ IsZero ((S r).homology n)) : |
| ¬ IsZero ((T).homology n) := |
| nonzero_of_functor_retract (HomologicalComplex.homologyFunctor (ModuleCat D) (.down ℕ) n) |
| (countTensorBarInclusion k D X N r) (countTensorBarProjection k D X N r) |
| (countTensorBar_retract k D X N r) h |
|
|
| /-- Direct input bridge for uniform bounds on the original count objects in |
| the integer derived category, yielding actual base-change Tor vanishing. -/ |
| theorem baseChangeTor_zero_of_counts (n : ℕ) |
| (h : ∀ r, IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) (-(n:ℤ))).obj |
| ((integerQ D).obj (S r)))) : |
| IsZero (((Balanced.tensorFunctor (fstHom k D X)).leftDerived n).obj (ModuleCat.of A N)) := |
| (tensorBar_homology_zero_of_counts k D X N n |
| (fun r => (h r).of_iso (integerQHomologyIso D (S r) n).symm)).of_iso |
| (Balanced.tensorBarHomologyIso (fstHom k D X) N n) |
|
|
| /-- A surviving original count homology is a nonzero actual Tor group. -/ |
| theorem baseChangeTor_nonzero_of_count (r n : ℕ) |
| (h : ¬ IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) (-(n:ℤ))).obj |
| ((integerQ D).obj (S r)))) : |
| ¬ IsZero (((Balanced.tensorFunctor (fstHom k D X)).leftDerived n).obj (ModuleCat.of A N)) := by |
| intro hz |
| have hc : IsZero ((S r).homology n) → False := by |
| intro hc |
| exact h (hc.of_iso (integerQHomologyIso D (S r) n)) |
| apply tensorBar_homology_nonzero_of_count k D X N r n hc |
| exact hz.of_iso (Balanced.tensorBarHomologyIso (fstHom k D X) N n).symm |
| end DerivedBar.ParityBridge |
|
|
| end |
|
|
| end OAI |
|
|
